| import Mathlib |
| import OAI.Computability.VertexCover.Analysis.StarMeanContinuous |
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| namespace OAI |
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| namespace VertexCover.LabelCover |
|
|
| noncomputable def starLocalKey (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (e : PositionPair d) (c : Fin Φ.M) : |
| Option (Fin Φ.M) × Option (Fin Φ.u) × Option (Fin Φ.v) := by |
| classical |
| exact (if e.1.1 = k ∨ e.1.2 = k then none else some c, |
| if e.1.1 = k then none else some (Φ.left c), |
| if e.1.2 = k then none else some (Φ.right c)) |
|
|
| theorem starKey_apply (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) (e : PositionPair d) : |
| Φ.starKey k seed e = Φ.starLocalKey k e (seed e) := rfl |
|
|
| noncomputable def starCell (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) (e : PositionPair d) : Finset (Fin Φ.M) := by |
| classical |
| exact Finset.univ.filter (fun c => Φ.starLocalKey k e c = Φ.starLocalKey k e (seed e)) |
|
|
| theorem mem_starCell (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) (e : PositionPair d) (c : Fin Φ.M) : |
| c ∈ Φ.starCell k seed e ↔ Φ.starLocalKey k e c = Φ.starLocalKey k e (seed e) := by |
| classical |
| simp [starCell] |
|
|
| theorem mem_starFiber_iff_cells (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed seed' : Φ.Seeds d) : |
| seed' ∈ Φ.starFiber k seed ↔ ∀ e, seed' e ∈ Φ.starCell k seed e := by |
| rw [Φ.mem_starFiber] |
| constructor |
| · intro h e |
| exact (Φ.mem_starCell k seed e _).mpr (congrFun h e) |
| · intro h |
| funext e |
| exact (Φ.mem_starCell k seed e _).mp (h e) |
|
|
| noncomputable def starFiberEquiv (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) : (Φ.starFiber k seed) ≃ (∀ e, Φ.starCell k seed e) where |
| toFun seed' e := ⟨seed'.1 e, (Φ.mem_starFiber_iff_cells k seed seed'.1).mp seed'.2 e⟩ |
| invFun choices := ⟨fun e => (choices e).1, |
| (Φ.mem_starFiber_iff_cells k seed _).mpr (fun e => (choices e).2)⟩ |
| left_inv seed' := by |
| apply Subtype.ext |
| rfl |
| right_inv choices := by |
| funext e |
| apply Subtype.ext |
| rfl |
|
|
| theorem starFiber_card_product (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) : |
| (Φ.starFiber k seed).card = ∏ e, (Φ.starCell k seed e).card := by |
| classical |
| have h := Fintype.card_congr (Φ.starFiberEquiv k seed) |
| simpa only [Fintype.card_coe, Fintype.card_pi] using h |
|
|
| theorem starCell_nonempty (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) (e : PositionPair d) : (Φ.starCell k seed e).Nonempty := |
| ⟨seed e, (Φ.mem_starCell k seed e _).mpr rfl⟩ |
|
|
| theorem finiteMean_starFiber_product (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) (f : Φ.Seeds d → ℝ) : |
| VertexCover.finiteMean (fun seed' : Φ.starFiber k seed => f seed'.1) = |
| VertexCover.finiteMean (fun choices : (∀ e, Φ.starCell k seed e) => |
| f (fun e => (choices e).1)) := by |
| classical |
| have hs := (Φ.starFiberEquiv k seed).sum_comp |
| (fun choices : (∀ e, Φ.starCell k seed e) => f (fun e => (choices e).1)) |
| have hc := Fintype.card_congr (Φ.starFiberEquiv k seed) |
| unfold VertexCover.finiteMean |
| rw [hc] |
| exact congrArg (fun x : ℝ => x / Fintype.card (∀ e, Φ.starCell k seed e)) hs |
|
|
| theorem starCell_nonincident (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) (e : PositionPair d) (hL : e.1.1 ≠ k) (hR : e.1.2 ≠ k) : |
| Φ.starCell k seed e = {seed e} := by |
| classical |
| ext c |
| rw [Φ.mem_starCell, Finset.mem_singleton] |
| constructor |
| · intro h |
| have hh := congrArg Prod.fst h |
| simpa only [starLocalKey, ite_eq_right (not_or.mpr ⟨hL, hR⟩), Option.some.injEq] using hh |
| · rintro rfl |
| rfl |
|
|
| theorem starCell_left (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) (e : PositionPair d) (hL : e.1.1 = k) : |
| Φ.starCell k seed e = Finset.univ.filter (fun c => Φ.right c = Φ.right (seed e)) := by |
| classical |
| have hR : e.1.2 ≠ k := by |
| intro h |
| exact (ne_of_lt e.2) (hL.trans h.symm) |
| ext c |
| simp [Φ.mem_starCell, starLocalKey, hL, hR] |
|
|
| theorem starCell_right (Φ : LabelCover) {d : ℕ} (k : Fin d) |
| (seed : Φ.Seeds d) (e : PositionPair d) (hR : e.1.2 = k) : |
| Φ.starCell k seed e = Finset.univ.filter (fun c => Φ.left c = Φ.left (seed e)) := by |
| classical |
| have hL : e.1.1 ≠ k := by |
| intro h |
| exact (ne_of_lt e.2) (h.trans hR.symm) |
| ext c |
| simp [Φ.mem_starCell, starLocalKey, hL, hR] |
|
|
| end VertexCover.LabelCover |
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| end |
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| end OAI |
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